Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 67624, 9 pages
doi:10.1155/JIA/2006/67624

Schur-convexity of the complete elementary symmetric function

Kaizhong Guan

 Department of Mathematics and Physics, Nanhua University, Hengyang, Hunan 421001, China
Received 2 October 2004; Revised 15 January 2005; Accepted 27 January 2005

Abstract

We prove that the complete elementary symmetric function cr=cr(x)=Cn[r](x)=∑i1+⋯+in=rx1i1⋯xnin and the function φr(x)=cr(x)/cr−1(x) are Schur-convex functions in R+n={(x1,x2,…,xn)|xi>0}, where i1,i2,…,in are nonnegative integers, r∈N={1,2,…}, i=1,2,…,n. For which, some inequalities are established by use of the theory of majorization. A problem given by K. V. Menon (Duke Mathematical Journal 35 (1968), 37–45) is also solved.